#7509386

order	512
nilpotency class	3
isoclinism	isoclinic class of #7500689
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	#2537	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2290 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1379 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 5 generators on 32 points:

(1,2)(3,4)(5,7)(6,8)(9,10)(11,14)(12,16)(13,15);(1,3)(2,6)(4,7)(5,8)(9,11,13,12)(10,14,15,16)(17,18)(19,27)(20,24)(21,30)(22,26)(23,28)(25,31)(29,32);(9,12,13,11)(10,14,15,16)(17,19,22,20)(18,23,26,32)(21,29,31,28)(24,30,27,25);(2,7)(3,8)(10,15)(14,16)(17,20,22,19)(18,24,26,27)(21,29,31,28)(23,30,32,25);(1,4,5,6)(2,3,7,8)(19,20)(23,32)(24,27)(28,29)

Polycyclic presentation:

2^9|b^2=g,c^2=gi,d^2=i,e^2=f,h^2=i,b^a=bf,c^a=cg,d^a=dfg,c^b=ch,d^b=df,h^b=hi,e^c=ei,h^c=hi,e^d=ei

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7416582);